Ergodic theory of low-dimensional dynamical systems

低维动力系统的遍历理论

• 批准号: RGPIN-2017-06521
• 负责人: Tiozzo, Giulio
• 金额: $ 3.64万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=749632
• 关键词: Ergodic; theory; low; dimensional; dynamical

The goal of this project is to investigate dynamical properties of low-dimensional systems using methods from ergodic theory. In particular, I will develop the theory of core entropy of polynomials as recently introduced by W. Thurston. Properties of this function will be related to the geometry and topology of the Mandelbrot set, and will be used to better understand the geometry of the parameter space of polynomials of higher degree and rational maps. Moreover, I will investigate the ergodic theory of random walks on groups acting on spaces with hyperbolic properties, and derive applications to geometric group theory, Teichmueller dynamics and low-dimensional topology. This will extend the theory of random walks on Lie groups, as developed by Furstenberg, Margulis, Zimmer, and others, to groups of interest in geometry and topology such as the mapping class group and the group of outer automorphisms of the free group.

该项目的目标是使用遍历理论的方法研究低维系统的动力学特性。特别是，我将发展 W. Thurston 最近介绍的多项式核心熵理论。该函数的性质将与Mandelbrot集的几何和拓扑相关，并且将用于更好地理解更高次多项式和有理映射的参数空间的几何。此外，我将研究作用于具有双曲性质的空间的群的随机游走的各态遍历理论，并推导其在几何群论、Teichmueller 动力学和低维拓扑中的应用。这将把由 Furstenberg、Margulis、Zimmer 等人开发的李群上的随机游走理论扩展到几何和拓扑中感兴趣的群，例如映射类群和自由群的外自同构群。